Limitations of perturbative techniques in the analysis of rhythms and oscillations

TL;DR

This paper critically examines the limitations of perturbative methods, especially the phase response curve, in analyzing oscillatory systems, highlighting scenarios where these methods fail to predict chaotic behavior or overestimate regularity.

Contribution

The paper identifies specific dynamical scenarios where perturbative techniques based on PRC are inaccurate, emphasizing the need for caution in their application.

Findings

PRC-based methods miss shear-induced chaos.

Sticky phase-space structures cause overestimation of oscillation regularity.

Simple neuron model demonstrates these phenomena.

Abstract

Perturbation theory is an important tool in the analysis of oscillators and their response to external stimuli. It is predicated on the assumption that the perturbations in question are "sufficiently weak", an assumption that is not always valid when perturbative methods are applied. In this paper, we identify a number of concrete dynamical scenarios in which a standard perturbative technique, based on the infinitesimal phase response curve (PRC), is shown to give different predictions than the full model. Shear-induced chaos, i.e., chaotic behavior that results from the amplification of small perturbations by underlying shear, is missed entirely by the PRC. We show also that the presence of "sticky" phase-space structures tend to cause perturbative techniques to overestimate the frequencies and regularity of the oscillations. The phenomena we describe can all be observed in a simple 2D neuron model, which we choose for illustration as the PRC is widely used in mathematical neuroscience.